A lobby group against gay marriage, formed by MPs and bishops, is embroiled in a row after one of its leaflets claimed that the logical argument for reforming the law would be equally applicable to the legalisation of incest and polygamy.

The eight-page document, produced by the Keep Marriage Special campaign, whose supporters include the former bishop of Rochester, the Rt Rev Michael Nazir Ali, warns of the "consequential impact" of the reform.

The glossy leaflet, which has a picture of a mixed-race couple in an embrace on the cover, claims: "If the only basis for marriage is the desire of the parties to get married then there is, according to the logic of this proposal, no reason not to open up marriage to more than just same-sex couples. Polygamy, polyandry and incest would all be permissible."

It adds: "The immigration service is already swamped with false marriages – this would only add to their problems."

At the moment, only men and women are permitted to marry; same-sex couples can only form civil partnerships, which became law in 2005. Civil partnerships give same-sex couples the right to the same legal treatment as married couples across a range of matters, but the law does not allow such unions to be referred to as marriages.

A Home Office consultation exercise on a change to the law is believed to have received more than 100,000 responses, including one from the Keep Marriage Special campaign, whose vice-presidents include the Democratic Unionist MPs Nigel Dodds and Jeffrey Donaldson and Bishop David Samuel, president of the Protestant Reformation Society. The campaign's president is Viscount Brentford, of the Church Society.

The campaign's leaflet boasts: "We include members both of the House of Lords and the House of Commons, as well as bishops, clergy and members of the Church of England and the free churches."

It adds: "There is nothing homophobic about stating the biblical position that all sexual activity outside marriage is wrong. Christians must show love to all people and are themselves sinners whose sins have been forgiven. They have a duty to tell the truth about right and wrong and to proclaim the good news that God forgives the sins of all who repent and trust in Jesus Christ."

However, Ethan Bourne, from the cross-party equal rights video campaign, Out4Marriage – which is backed by the home secretary, Theresa May, and the deputy prime minister, Nick Clegg – described the Keep Marriage Special campaign as "extremist and confused".

He added: "We are not for incest or polygamy, but for allowing marriage between two people of the same sex who love each other. What the Keep Marriage Special campaign is doing is talking about a different issue altogether."

Although David Cameron supports the change, a number of Conservative MPs have voiced their opposition to the measures. They include the defence secretary, Philip Hammond, who has said that gay marriage is "not a priority", and the Northern Ireland secretary, Owen Paterson, who has said he will not vote for it.

However, Clegg issued a stark warning last week to Conservative MPs and church leaders who oppose gay marriage. The deputy prime minister, who last week became the most senior politician to record a video message for the Out4Marriage group, said: "I've always been very clear on this: love is the same, straight or gay, so the civil institution should be the same too. All couples should be able to make that commitment to one another, regardless of who they love."

A running survey of MPs' stated views by the Coalition for Equal Marriage suggests 245 MPs are likely to support legalising gay marriage, with 62 against and 16 neutral. The views of the remaining 327 MPs are not yet known.